Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
586 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 586 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#99 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×C22⋊C4 | C23×C53 |
| prime radical | C25×C53 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C24×C53 ◁ C22×C22⋊C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C53×#1351 | C2 | semidirect |
| 5. C53×#1344 | C2 | semidirect |
| 6. C2×C22⋊Q8×C53 | C4 | semidirect |
| 10. C53×#397 | C4 | semidirect |
| 12. C53×#393 | C4 | semidirect |
| 18. C4×C22⋊C4×C53 | C4 | semidirect |
| 24. C22×Q8×C53 | C2×C4 | semidirect |
| 32. C2×C4⋊C4×C53 | C2×C4 | semidirect |
| 56. #95×C53 | C2×C4 | semidirect |
| 57. #95×C53 | C2×C4 | semidirect |
| 167. #11759 | C53 | direct (N × H) |
| 209. #1351 | C2×C53 | semidirect |
| 210. #1344 | C2×C53 | semidirect |
| 221. C2×C22⋊Q8 | C4×C53 | semidirect |
| 225. #397 | C4×C53 | semidirect |
| 227. #393 | C4×C53 | semidirect |
| 233. C4×C22⋊C4 | C4×C53 | semidirect |
| 238. C53 | #11759 | direct (N × H) |
| 240. C22×Q8 | C2×C4×C53 | semidirect |
| 248. C2×C4⋊C4 | C2×C4×C53 | semidirect |
| 272. #95 | C2×C4×C53 | semidirect |
| 273. #95 | C2×C4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2561 | 2 |
| C2×C53×#342 | 2 |
| C53×#1462 | 2 |
| C53×#1351 | 2 |
| C53×#1344 | 2 |
| #11759 | 53 |